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THE PHYSICAL MEANING OF PHASE COEFFICIENTS
The question of what information about an asteroid's surface is contained in a measurement of the phase coefficient between phase angles of 10° and 30° is examined in detail. Contrary to some past claims it is shown that absolute reflectivities cannot be derived from phase coefficients. Furthermore, typical asteroid phase coefficients cannot be interpreted unambiguously. This is because the observed phase coefficient may depend as much on the photometric properties of an individual surface element as on the degree of large-scale surface roughness, and these two effects are impossible to separate if only disk integrated measurements are available. The wavelength dependence of asteroid phase coefficients should be small and should contain little information about the surface. In the case of irregular asteroids with macroscopically rough surfaces, the importance of large-scale shadowing, and hence the observed phase coefficient, will depend on the aspect of the asteroid. In such cases, therefore, phase coefficients must be carefully defined to be meaningful. It should be possible, in some cases, to estimate the relative surface roughness of two quasi-spherical asteroids by combining photometric and polarimetric observations. For example, if the two asteroids have almost identical polarization curves but quite different phase coefficients, it is likely that the asteroid with the larger phase coefficient has a macroscopically rougher surface.
One of the aims of asteroid photometry is to obtain information about the physical characteristics, such as texture, composition, and large-scale roughness, of asteroid surfaces. In this paper I wish to concentrate on a single aspect of asteroid photometry and consider in detail what information can be derived from observed phase coefficients. For instance, is it possible, as Bell (1917), Stumpff (1948), Widorn (1964), and recently Gehrels et al. (1970) have tried to do, to determine the absolute reflectivities of asteroids in this way?
I will use the term “phase coefficient” in a restricted sense. From Earth, few asteroids can be observed at phase angles larger than 30°. Also, at very small phase angles an additional surge in brightness (the “opposition effect”) is usually present (Gehrels, 1956, 1967). The details of this opposition surge contain important information about the surface texture (Hapke, 1963; Irvine, 1966), but few asteroids have been observed at sufficiently small phase angles to determine accurately this part of their phase curves. I will therefore use the term “phase coefficient” to mean the slope (in magnitudes per degree of phase) of the observed phase curve between 10° and 30°. The problem of understanding the physical implications of this quantity (which I will denote by 3) can be divided into two parts:
(1) To adequately describe the scattering properties of an individual small element of the surface of a typical asteroid
(2) To determine what additional effects are introduced by shadowing due to large-scale roughness
These two questions are dealt with in turn in the next two sections.
THE SCATTERING PROPERTIES OF A SMALL SURFACE ELEMENT
Observational evidence suggests that the surface of a typical asteroid is similar to that of the Moon; that is, microscopically rough and intricate, and made up largely of a dark material in which multiple scattering is not dominant. The scattering properties of such surfaces have been considered by Irvine (1966); his model gives an exact treatment of the scattering properties of a dark, particulate layer under the following assumptions:
(1) All particles are spherical and of uniform radius.
(2) The particles are large enough that shadowing can be dealt with in terms of geometric optics.
(3) The particles are dark enough for multiple scattering to be negligible.
When a parallel beam of light is incident on an element of such a surface, at an angle i, the specific intensity I of the light scattered at an angle e (making a phase angle a with the incident direction) is given by
COS l I(i, e, o] or [*] — S(i, e, o, D) (1) COS i + COS e where ôjo = scattering albedo of a single particle q}(0) = phase function of a single particle S(i, e, of D) = Irvine shadowing function for the surface
The parameter D is related to the compaction of the surface as follows. If p is the mean density of a macroscopic volume element of the surface, and po is
the mean density of a single particle, then
For uniform, equally hard spheres, D cannot exceed 0.176 (Beresford, 1969). For the Moon's top surface, Hapke (1963) estimates plpo = 0.1, which corresponds to D = 0.024. Using the equations given by Irvine, it is easy to show that S(i, e, or; D) does not depend strongly on either i or e individually, so that
Therefore equation (1) may be rewritten as
* cos i I(i, e, a) - do — f(o, D) (2) COS i + COS e where f(o, D) = S(o, D) q}(0). Although this equation is based on simplifying assumptions, it does adequately represent laboratory measurements on dark, microscopically rough surfaces. Furthermore, it holds even for surfaces in which the individual particles are not physically separate but are fused together as, for example, in furnace slag. The validity of equation (2) can be easily tested for any surface in the laboratory by making measurements of I(i, e, o] as a function of o at a series of fixed values of e, say at e = 0°, 30°, and 60°. From each set of measurements corresponding to a given e, an empirical f(or, D) can be determined using equation (2). If this equation is applicable to the surface, all the f(o, D) values so obtained will be identical. Such a test is carried out, using measurements on a sample of dark furnace slag (Halajian, 1965), in figure 1 where all the f(o, D) values have been normalized to unity at a = 10°. Because a single f(o, D) is indicated, equation (2) appears to be valid for this surface, even though this surface is not “particulate” in the usual sense. This test can be carried out with equal success for dark surfaces which are particulate in the normal sense. In fact, Halajian
Figure 1.-The fifunction for a layer of dark furnace slag, from measurements in V by Halajian (1965). This sample has a normal reflectivity of 0.09 and photometric properties very similar to those of the lunar surface. The function is normalized to unity at a = 10°.
(1965) found that many particulate surfaces (volcanic cinders, for example) have values of f(o, D) almost identical to that shown in figure 1, which incidentally, is very similar to that of the lunar surface. I will now show that the f(o, D) shown in figure 1 can be adequately reproduced using the Irvine model. In doing this, it is convenient to choose for q}(0), the one parameter family of single particle phase functions introduced by Henyey and Greenstein (1941), ‘pHC(a, G) 1 – G2 (3) hg(0. (1 + G2 – 2G coso)3/2
The parameter G = < coso X describes the nature of the scattering. For G = +1, there is complete backscattering; for G = -1, complete forwardscattering; and for G = 0 the scattering is isotropic. The measured f(o., D) shown in figure 1 can only be matched for a very small range of G (0.30 to 0.35) (fig. 2). This indicates that effectively the individual particles are slightly backscattering, a result to be expected for large, opaque particles with rough surfaces. In figure 2, a reasonable choice of D = 0.03 is used, but the conclusions do not depend strongly on the value of D.
Figure 2.-Comparison of the fifunction of figure 1 with two theoretical predictions using the Irvine model and a Henyey-Greenstein phase function. The points represent the mean values of f at each phase angle, taken from figure 1.
I conclude that the Irvine model is adequate for describing the scattering properties of dark, microscopically intricate surfaces. Furthermore, it seems immaterial whether the particles of the surface are physically free or fused together.
THE EFFECTS OF LARGE-SCALE ROUGHNESS: MACROSCOPIC
Unfortunately, the general problem of shadowing on a randomly rough two-dimensional surface is extremely complicated. Ideally, one wishes to know for each angle of illumination and each angle of observation what parts of the surface are both illuminated and seen. The surface can be specified statistically in terms of the height deviations from an arbitrary mean level or in terms of the distribution of surface slopes. So far, solutions exist only for onedimensional surfaces (for example Beckmann, 1965, and Saunders, 1967), and I will therefore use a contrived, but convenient model, first introduced by Hämeen-Anttila et al. (1965). In this model the surface is assumed to be bounded on top by a plane that is punctured by countless paraboloidal craters, whose axes of revolution are perpendicular to the plane. The shape of a crater is determined by the parameter Q = H/R, where H is the crater depth and R is its radius at the top level.
To study the effects of large-scale shadowing on the photometric properties of asteroids, it is convenient to first consider a model planet that is spherical and completely covered with paraboloidal craters of shape Q. (It is assumed that the craters do not overlap.) As Q increases from zero, so does the roughness of the model planet. The rms slope of such a surface is given by
and Q is related to the maximum surface slope by the relation
Q = tan "max (5) 2
For 6max <35°, for example, Qs 0.35.
It is implicitly assumed in the model that, on the one hand, the number of craters per resolution element is very large, whereas on the other hand, each crater is large enough to contain a large number of individual scattering elements. Also, the surface reflectivity is assumed to be low enough that shadows are not affected by multiple scattering.
To determine the total amount of light j(o) scattered by the model planet toward Earth at a phase angle a, an integration over the illuminated part of the disk must be performed:
where cos e do is the projected area of the surface element do, and T is the effective specific intensity of the light scattered by that element toward Earth. Numerically, this process is conveniently carried out by the method of Horak (1950) in which the integration is replaced by a weighted sum over a grid of points covering the illuminated part of the disk. At each point of this grid, I is